Pythagoras theorem


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Pythagoras theorem

  1. 1. Pythagoras of Samos (Ancient Greek: Πυθαγόρας ὁ Σάμιος [Πυθαγόρης in Ionian Greek] Pythagóras ho Sámios "Pythagoras the Samian", or simply Πυθαγόρας; b. about 570 – d. about 495 BC)[1][2] was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him. He was born on the island of Samos, and might have travelled widely in his youth, visiting Egypt and other places seeking knowledge. Around 530 BC, he moved to Croton, a Greek colony in southern Italy, and there set up a religious sect. His followers pursued the religious rites and practices developed by Pythagoras, and studied his philosophical theories. The society took an active role in the politics of Croton, but this eventually led to their downfall. The Pythagorean meeting-places were burned, and Pythagoras was forced to flee the city. He is said to have ended his days in Metapontum.
  2. 2. Pythagoras made influential contributions to philosophy and religious teaching in the late 6th century BC. He is often revered as a great mathematician, mystic and scientist, but he is best known for the Pythagorean theorem which bears his name. However, because legend and obfuscation cloud his work even more than that of the other pre-Socratic philosophers, one can give only a tentative account of his teachings, and some have questioned whether he contributed much to mathematics and natural philosophy. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues and successors. Whether or not his disciples believed that everything was related to mathematics and that numbers were the ultimate reality is unknown. It was said that he was the first man to call himself a philosopher, or lover of wisdom,[3] and Pythagorean ideas exercised a marked influence on Plato, and through him, all of Western philosophy.
  3. 3. Pythagoras made many contributions to Mathematics and Physics but the contribution that we will be exploring is: The Pythagorean Theorem. The Pythagorean Theorem is a formula that can be used only when working with right triangles. It can help you find the length of any side of a right triangle. Pythagoras wasn't the first to discover this formula. The Babylonians and the Chinese worked with this concept years before Pythagoras. Pythagoras gets most of the credit for it though because he was the first to prove why it works. Several other proofs came about after his.
  4. 4. The Pythagorean Theorem was one of the earliest theorems known to ancient civilizations. This famous theorem is named for the Greek mathematician and philosopher, Pythagoras. Pythagoras founded the Pythagorean School of Mathematics in Cortona, a Greek seaport in Southern Italy . He is credited with many contributions to mathematics although some of them may have actually been the work of his students. The Pythagorean Theorem is Pythagoras' most famous mathematical contribution. According to legend, Pythagoras was so happy when he discovered the theorem that he offered a sacrifice of oxen. The later discovery that the square root of 2 is irrational and therefore, cannot be expressed as a ratio of two integers, greatly troubled Pythagoras and his followers. They were devout in their belief that any two lengths were integral multiples of some unit length. Many attempts were made to suppress the knowledge that the square root of 2 is irrational. It is even said that the man who divulged the secret was drowned at sea. "The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides."
  5. 5. Reminder of square numbers: 12 = 1x1= 1 22 = 2x2= 4 32 = 3x3= 9 42 = 4x4= 16 Index number 32 Base number The index number tells us how many times the base number is multiplied by itself. e.g. 34 means 3 x 3 x 3 x 3 = 81 1,4,9,16, …. are the answers to a number being squared so they are called square numbers.
  6. 6. "Pythagoras' Theorem" can be written in one short equation: a2 + b2 = c2 Definition The longest side of the triangle is called the "hypotenuse", so the formal definition is: In a right angled triangle: the square of the hypotenuse is equal to the sum of the squares of the other two sides. Note: c is the longest side of the triangle a and b are the other two sides
  7. 7. Sure ... ? Let's see if it really works using an example. Example: A "3,4,5" triangle has a right angle in it. Let's check if the areas are the same: 32 + 42 = 52 Calculating this becomes: 9 + 16 = 25 It works ... like Magic!
  8. 8. Why Is This Useful? If we know the lengths of two sides of a right angled triangle, we can find the length of the third side. (But remember it only works on right angled triangles!) It works the other way around, too: when the three sides of a triangle make a2 + b2 = c2, then the triangle is right angled. Example: Does this triangle have a Right Angle? Does a2 + b2 = c2 ? a2 + b2 = 102 + 242 = 100 + 576 = 676 c2 = 262 = 676 They are equal, so ... Yes, it does have a Right Angle! Example: Does an 8, 15, 16 triangle have a Right Angle? Does 82 + 152 = 162 ? 82 + 152 = 64 + 225 = 289, but 162 = 256 So, NO, it does not have a Right Angle
  9. 9. means think what is multiplied by itself to make this number? Square root Answer these questions: 1 1 4 2 9 3 16 4 49 7 Use your calculator to answer these questions: 5 .8 2.408 25 . 4 5.040 169 13 400 20 1 21 11 1 00 10 31.623 8100 81 1000 90 225 15 361 19
  10. 10. Cut the squares away from the right angle triangle and cut up the segments of square ‘a’ q To show how this works: b Draw line segment , parallel with the hypotenuse of the triangle a x Draw line segment pq, at right angles to Line segment xy. y p
  11. 11. Now rearrange them to look like this. You can see that they make a square with length of side ‘c’. This demonstrates that the areas of squares a and b add up to be the area of square c a +b 2 2 =c 2
  12. 12. x 1 3 cm x 3 2 2 4 2 4 2 x 3 x x 4 cm 9 + 16 25 x 2 2 5 cm x x 2 5 2 2 12 12 cm 5 x 5 cm x 169 x 13 cm 12 2 2
  13. 13. x x 2 11 2 9 5 x 2 9 11m x xm 11 x 23.8 cm 2 xm 7 x x 3.4 cm x 7.1 cm x cm x 2 11 2 11 23.8 2 2 21.1 cm (1 dp) 2 7.1 2 3.4 2 2 3.4 2 7.1 7 2 25 7 24 m 2 25 m 23.8 x 11 cm 2 xm 6.3 m (1 dp) 9m 6 8 2 25 2 x x 2 Now do these: 2 7.9 cm (1 dp) 7m
  14. 14. The first proof begins with a rectangle divided up into three triangles, each of which contains a right angle. This proof can be seen through the use of computer technology, or with something as simple as a 3x5 index card cut up into right triangles .
  15. 15. The next proof is another proof of the Pythagorean Theorem that begins with a rectangle. It begins by constructing rectangle CADE with BA = DA. Next, we construct the angle bisector of <BAD and let it intersect ED at point F. Thus, <BAF is congruent to <DAF, AF = AF, and BA = DA. So, by SAS, triangle BAF = triangle DAF. Since <ADF is a right angle, <ABF is also a right angle. Next, since m<EBF + m<ABC + m<ABF = 180 degrees and m<ABF = 90 degrees, <EBF and <ABC are complementary. Thus, m<EBF + m<ABC = 90 degrees. We also know that m<BAC + m<ABC + m<ACB = 180 degrees. Since m<ACB = 90 degrees, m<BAC + m<ABC = 90 degrees. Therefore, m<EBF + m<ABC = m<BAC + m<ABC and m<BAC = m<EBF. By the AA similarity theorem, triangle EBF is similar to triangle CAB.
  16. 16. Now, let k be the similarity ratio between triangles EBF and CAB. . Thus, triangle EBF has sides with lengths ka, kb, and kc. Since FB = FD, FD = kc. Also, since the opposite sides of a rectangle are congruent, b = ka + kc and c = a + kb. By solving for k, we have Thus, By cross-multiplication, Therefore, and we have completed the proof.
  17. 17. The next proof of the Pythagorean Theorem that will be presented is one that begins with a right triangle. In the next figure, triangle ABC is a right triangle. Its right angle is angle C. Next, draw CD perpendicular to AB as shown in the next figure. From Figures with CD, we have that (p + q) = c. By substitution, we get
  18. 18. The next proof of the Pythagorean Theorem that will be presented is one in which a trapezoid will be used. By the construction that was used to form this trapezoid, all 6 of the triangles contained in this trapezoid are right triangles. Thus, Area of Trapezoid = The Sum of the areas of the 6 Triangles And by using the respective formulas for area, we get: We have completed the proof of the Pythagorean Theorem using the trapezoid.
  19. 19. A boat sails due East from a Harbour (H), to a marker buoy (B),15 miles away. At B the boat turns due South and sails for 6.4 miles to a Lighthouse (L). It then returns to harbour. What is the total distance travelled by the boat? 15 miles H LH LH LH 2 15 2 15 2 6.4 B 2 6.4 6.4 miles 2 16.3 miles Total distance travelled = 21.4 + 16.4 = 37.7 miles L
  20. 20. A 12 ft ladder rests against the side of a house. The top of the ladder is 9.5 ft from the floor. How far is the base of the ladder from the house? L 2 12 L 2 12 L 2 9.5 2 9.5 2 12 ft 9.5 ft 7.3ft L
  21. 21. Finding the shortest distance
  22. 22. Ladder to reach the window
  23. 23. The amount of area represented by the triangles is the same for both the left and right sides of the figure. Take away the triangles. Then the area of the large square must equal the area of the two small squares. This proves the Pythagorean Theorem.